where we used the normalization conditions \({e}_{i}\cdot {e}_{i}=1\) , which implies \({e}_{i}\cdot \partial _{j}{e}_{i}=0\) . The final expression on the right of Eq. (REF ) can be recognised at once as the Berry connection [1]}. An analogous quantity appears in quantum theory, and that tells us how to transport a quantum mechanical state vector \(|\psi \rangle \)  around a space of parameters \(x_i\) . For our tangent vector \(|\psi \rangle \) , the curl of the coresponding Berry connection is—via Eq. (REF )—simply the Gaussian curvature of the surface.
